Page 292 - Aerodynamics for Engineering Students

274 Aerodynamics for Engineering Students

6.1 Introduction

In previous chapters the study of aerodynamics has been almost exclusively restricted
to incompressible flow. This theoretical model is really only suitable for the aero-
dynamics of low-speed flight and similar applications. For incompressible flow the
air density and temperature are assumed to be invariant throughout the flow field.
But as flight speeds rise, greater pressure changes are generated, leading to the
compression of fluid elements, causing in turn a rise in internal energy and, in
consequence, temperature. The resulting variation of these flow variables throughout
the flow field makes the results obtained from incompressible flow theory less and
less accurate as flow speeds rise. For example, in Section 2.3.4 we showed how use of
the incompressibility assumption led to errors in estimating the stagnation-pressure
coefficient of 2% at M = 0.3, rising to 6% at M = 0.5, and 28% at M = 1.
But these errors in estimating pressures and other flow variables are not the most
important disadvantage of using the incompressible flow model. Far more significant
is the marked qualitative changes to the flow field that take place when the local flow
speed exceeds the speed of sound. The formation of shock waves is a particularly
important phenomenon and is a consequence of the propagation of sound through
the air. In incompressible flow the fluid elements are not permitted to change in
volume as they pass through the flow field. And, since sound waves propagate by
alternately compressing and expanding the medium (see Section 1.2.7), this is tanta-
mount to assuming an infinite speed of sound. This has important consequences
when a body like a wing moves through the air otherwise at rest (or, equivalently,
a uniform flow of air approaches the body). The presence of the body is signalled by
sound waves propagating in all directions. If the speed of sound is infinite the
presence of the body is instantly propagated to the farthest extent of the flow field
and the flow instantly begins to adjust to the presence of the body.
The consequences of a finite speed of sound for the flow field are illustrated in
Fig. 6.11(p.308). Figure 6.11b depicts the wave pattern generated when a source of
disturbances (e.g. part of a wing) moves at subsonic speed into still air. It can be seen that
the wave fronts are closer together in the direction of flight. But, otherwise, the flow field is
qualitatively little different from the one (analogous to incompressible flow) correspond-
ing to the stationary source shown in Fig. 6.1 la. In both cases the sound waves eventually
reach all parts of the flow field (instantly in the case of incompressible flow). Contrast this
with the case, depicted in Fig. 6.1 IC, of a souce moving at supersonic speed. Now the
waves propagating in the forward direction line up to make planar wave fronts. The flow
field remains undisturbed outside the regions reached by these planar wave fronts, and
waves no longer propagate to all parts of the flow field. These planar wave fronts are
formed from a superposition of many sound waves and are therefore much stronger than
an individual sound wave. In many cases they correspond to shock waves, across which
the flow variables change almost discontinuously. At supersonic speeds the flow field is
fundamentally wavelike in character, meaning that information is propagated from one
part of the flow field to another along wave fronts. Whereas in subsonic flow fields, which
are not wavelike in character, information is propagated to all parts of the flow field.
This wavelike character of supersonic flow fields makes them qualitiatively different
from the low-speed flow fields studied in previous chapters. Furthermore, the existence
of shock waves brings about additional drag and many other undesirable changes from
the viewpoint of wing aerodynamics. As a consequence, the effects of flow compressi-
bility has a strong influence on wing design for high-speed flight even at subsonic flight
speeds. It might at first be assumed that shock waves only affect wing aerodynamics at